# Particle Velocity Calculator (Maxwell–Boltzmann)

URL: https://speedycalculator.com/calculators/physics/particles-velocity/
Last updated: 2026-10-30

> The particle velocity calculator finds the most probable, mean and root-mean-square speeds of gas particles of a given mass at a given temperature from the Maxwell–Boltzmann distribution.

## Inputs

- Particle mass (u (g/mol), kg, mₑ)
- Temperature (K, °C, °F)

## Results

- Most probable speed
- Mean speed
- Root-mean-square speed
- Mean kinetic energy

## Formula

```
v_p = √(2k_BT/m),  v̄ = √(8k_BT/(πm)),  v_rms = √(3k_BT/m)
```

## Explanation

Gas molecules zip around at hundreds of metres per second, with a spread of speeds described by the Maxwell–Boltzmann distribution. Lighter particles move faster at the same temperature – why hydrogen escapes from Earth's atmosphere.

### How to use the Particle Velocity Calculator

- **Particle mass**
- **Temperature**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the particle mass (in atomic mass units, molar mass or kg) and the temperature.

### Tips

- Use absolute temperature (K) in gas laws and efficiency formulas.
- Temperature differences are the same in K and °C.
- Heat = mass × specific heat × temperature change when no phase change occurs.
- Phase changes need latent heat at a constant temperature.

### Common mistakes to avoid

- Using °C in pV = nRT.
- Forgetting latent heat during melting or boiling.
- Confusing heat capacity (J/K) with specific heat (J/(kg·K)).

### Inputs explained

FieldWhat to enterParticle massUnits: u (g/mol), kg, mₑTemperatureUnits: K, °C, °F

### What each result means

- **Most probable speed** = sqrt(2×1.380649e-23×[Temperature]÷[Particle mass])
- **Mean speed** = sqrt(8×1.380649e-23×[Temperature]÷(pi×[Particle mass]))
- **Root-mean-square speed** = sqrt(3×1.380649e-23×[Temperature]÷[Particle mass])
- **Mean kinetic energy** = 1.5×1.380649e-23×[Temperature]÷1.602176634e-19×1000

### Example values

How the main result changes when **Particle mass** changes and everything else stays at the example values above:

Particle mass (u (g/mol))Root-mean-square speed (m/s)14.01722.520421.01589.935428.01510.899135.02456.96242.02417.147356.03361.2602

## Worked example

Inputs — Particle mass: 28.014 u (g/mol); Temperature: 293.15 K.

Results — Most probable speed = 417.14734 m/s; Mean speed = 470.70037 m/s; **Root-mean-square speed = 510.89906 m/s**; Mean kinetic energy = 37.89257 meV.

## Key terms

- **Maxwell–Boltzmann distribution**: Speed distribution of particles in a gas.

## FAQ

**Why are there three different speeds?**
The distribution is skewed: v_p < v̄ < v_rms.

**How fast are nitrogen molecules at room temperature?**
About 510 m/s (rms).

**Does mass affect average kinetic energy?**
No – it depends only on temperature.

**Why do light gases escape planets?**
Their speeds can exceed the escape velocity.

**What formula does the Particle Velocity Calculator use?**
v_p = √(2k_BT/m),  v̄ = √(8k_BT/(πm)),  v_rms = √(3k_BT/m).


## Sources

- [OpenStax – University Physics Volume 2](https://openstax.org/details/books/university-physics-volume-2)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)